LIMITING JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2008 ◽  
Vol 18 (10) ◽  
pp. 3175-3181 ◽  
Author(s):  
MARK MORABITO ◽  
ROBERT L. DEVANEY
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Julia Set ◽  
Singularly Perturbed ◽  
Parameter Plane ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Open Set ◽  
The Family

In this paper, we consider the family of rational maps given by [Formula: see text] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere.

scholarly journals GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

2012 ◽  
Vol 22 (12) ◽  
pp. 1250301 ◽  
Author(s):  
SUZANNE HRUSKA BOYD ◽  
MICHAEL J. SCHULZ
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Julia Sets ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Geometric Limit ◽  
The Family ◽  
Mandelbrot Sets

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

scholarly journals Real analyticity of Hausdorff dimension for expanding rational semigroups

2009 ◽  
Vol 30 (2) ◽  
pp. 601-633 ◽  
Author(s):  
HIROKI SUMI ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Dimension Function ◽  
Analytic Family ◽  
Open Set ◽  
Extensive Collection ◽  
Real Analytic ◽  
Parameter Function

AbstractWe consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the Hausdorff dimension of the Julia set is a real-analytic and plurisubharmonic function of the parameter. Moreover, we provide an extensive collection of examples of analytic families of semigroups satisfying all of the above conditions and we analyze in detail the corresponding Bowen’s parameters and Hausdorff dimension function.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

scholarly journals Some Results and Problems Concerning Chordal Principal Cluster Sets

1967 ◽  
Vol 29 ◽  
pp. 7-18 ◽  
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Continuous Curve ◽  
Cluster Set ◽  
Asymptotic Values ◽  
Principal Cluster

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).

scholarly journals THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

2002 ◽  
Vol 85 (2) ◽  
pp. 467-492 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Julia Set ◽  
Density Theorem ◽  
Finite Union ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Open Set ◽  
Almost All ◽  
Scenery Flow ◽  
Measure Of Maximal Entropy

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

Meromorphic multifunctions in complex dynamics

1992 ◽  
Vol 12 (1) ◽  
pp. 39-52 ◽  
Author(s):  
L. Baribeau ◽  
T. J. Ransford
Keyword(s):  
Critical Points ◽  
Complex Dynamics ◽  
Julia Set ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Analytic Family ◽  
Limit Sets ◽  
Upper Semicontinuous ◽  
Open Set ◽  
Similar Vein

AbstractLet {RA} be an analytic family of rational maps and denote by j(λ) the Julia set of Rλ. We prove that the upper semicontinuous regularization j(λ) of j(λ) (which coincides with j(λ) for all λ in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.

An exceptional set in the ergodic theory of rational maps of the Riemann sphere

1997 ◽  
Vol 17 (2) ◽  
pp. 253-267 ◽  
Author(s):  
A. G. ABERCROMBIE ◽  
R. NAIR
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Hausdorff Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Exceptional Set ◽  
Rational Map ◽  
Pointwise Ergodic Theorem ◽  
Conformal Measure

A rational map $T$ of degree not less than two is known to preserve a measure, called the conformal measure, equivalent to the Hausdorff measure of the same dimension as its Julia set $J$ and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoff's pointwise ergodic theorem almost every $z$ in $J$ with respect to the conformal measure has an orbit that is asymptotically distributed on $J$ with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set $\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ of degree at least two we set \[ S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}. \] We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$ and $J$ are equal.

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